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Mademuasel [1]
3 years ago
13

Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.

Mathematics
1 answer:
grandymaker [24]3 years ago
7 0
(2x+1)^{\cot x}=\exp\left(\ln(2x+1)^{\cot x}\right)=\exp\left(\cot x\ln(2x+1)\right)=\exp\left(\dfrac{\ln(2x+1)}{\tan x}\right)

where \exp(x)\equiv e^x.

By continuity of e^x, you have

\displaystyle\lim_{x\to0^+}\exp\left(\dfrac{\ln(2x+1)}{\tan x}\right)=\exp\left(\lim_{x\to0^+}\dfrac{\ln(2x+1)}{\tan x}\right)

As x\to0^+ in the numerator, you approach \ln1=0; in the denominator, you approach \tan0=0. So you have an indeterminate form \dfrac00. Provided the limit indeed exists, L'Hopital's rule can be used.

\displaystyle\exp\left(\lim_{x\to0^+}\dfrac{\ln(2x+1)}{\tan x}\right)=\exp\left(\lim_{x\to0^+}\dfrac{\frac2{2x+1}}{\sec^2x}\right)

Now the numerator approaches \dfrac21=2, while the denominator approaches \sec^20=1, suggesting the limit above is 2. This means

\displaystyle\lim_{x\to0^+}(2x+1)^{\cot x}=\exp(2)=e^2
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g100num [7]

Answer:

44580 inches

Step-by-step explanation:

Given, D (diameter)= 22 inches

Therefore, r (radius)= 22/2= 11 inches

Volume= \frac{4}{3}×π×r³

            =\frac{4}{3}×3.14×22×22×22

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3 0
3 years ago
Alicia's MP3 player contains 1,260 songs. Given that 35% of the songs are rock songs and 20% of the songs are rap songs, how man
ivanzaharov [21]
100% is a whole...

35+20=55
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4 0
3 years ago
Assume that y varies directly with x. If y = -5 when
steposvetlana [31]

Answer:

Thus, the value of x = -36 when y = 15

Step-by-step explanation:

We know that if y varies directly with x, we can express the relationship such as

y ∝ x

y = kx

k = y/x

where 'k' is called constant of variation.

Given

y = -5

x = 12

Using the equation

k = y/x = -5/12

Thus, the value of k = -5/12

Finding x when y = 15

y = 15

k = -5/12

substituting y = 15 and k = -5/12 in the equation

y = kx

15 = -5/12 (x)

15×12 = -5x

180 = -5x

divding both sides by -5

-5x/-5 = 180/-5

x = -36

Thus, the value of x = -36 when y = 15

8 0
3 years ago
Solve by completing the square<br>3x2 + x - 2 = 0​
DIA [1.3K]

Answer:

x = 2/3 or x = -1

Step-by-step explanation by completing the square:

Solve for x:

3 x^2 + x - 2 = 0

Divide both sides by 3:

x^2 + x/3 - 2/3 = 0

Add 2/3 to both sides:

x^2 + x/3 = 2/3

Add 1/36 to both sides:

x^2 + x/3 + 1/36 = 25/36

Write the left hand side as a square:

(x + 1/6)^2 = 25/36

Take the square root of both sides:

x + 1/6 = 5/6 or x + 1/6 = -5/6

Subtract 1/6 from both sides:

x = 2/3 or x + 1/6 = -5/6

Subtract 1/6 from both sides:

Answer:  x = 2/3 or x = -1

3 0
3 years ago
Ill GIVE BRAINELST AND EXTRA POINTS​
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Answer:

I believe the answer is (-7.4) Not really great at math either

Step-by-step explanation:

4 0
3 years ago
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